There is no pattern. Or rather, the symbolic engine uses the next one that it has not used before in the session, including using it internally during computations that are not shown to you. Calculations will not necessarily get done in any particular order, either.
You can use symvar() on the expressions before and after to find the names that have been introduced, to give you a list of constants of integration without context in the expression. You could pull the expression apart if you really wanted to, or you could just rename them without caring where they occur or what they mean.
If you want the new names to have meaning reflecting what they are for, you should probably be using initial conditions; it is completely allowed to use symbolic initial conditions such as x(0)==x0 and the computation engine will use that information. Caution, though: it is not uncommon that when you specify an initial condition, the computation engine ends up discarding a valid computation path and is not able to find a solution when one exists. It is also common for pretty strange computations to show up, like x being a function of tau involving a constant omega19 where omega19 is the root of a integral expression involving a function that itself involves the root of an integral expression...
